Edexcel S2 (Statistics 2) 2014 June

1. Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.
   1. Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
      1. exactly 7
      2. at least 10
   A patient arrives at 11.30 a.m.
2. Find the probability that the next patient arrives before 11.45 a.m.

2. The length of time, in minutes, that a customer queues in a Post Office is a random variable, \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } c \left( 81 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 9
0 & \text { otherwise } \end{array} \right.$$ where \(c\) is a constant.

1. Show that the value of \(c\) is \(\frac { 1 } { 486 }\)
2. Show that the cumulative distribution function \(\mathrm { F } ( t )\) is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c c } 0 & t < 0
   \frac { t } { 6 } - \frac { t ^ { 3 } } { 1458 } & 0 \leqslant t \leqslant 9
   1 & t > 9 \end{array} \right.$$
3. Find the probability that a customer will queue for longer than 3 minutes. A customer has been queueing for 3 minutes.
4. Find the probability that this customer will be queueing for at least 7 minutes. Three customers are selected at random.
5. Find the probability that exactly 2 of them had to queue for longer than 3 minutes.

1. A company claims that it receives emails at a mean rate of 2 every 5 minutes.
   1. Give two reasons why a Poisson distribution could be a suitable model for the number of emails received.
   2. Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4 . The probability of rejection in each tail should be as close as possible to 0.025
   3. Find the actual level of significance of this test.
   To test this claim, the number of emails received in a random 10 minute period was recorded. During this period 8 emails were received.
2. Comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the Wimbledon Men's Tennis Tournament final, 2 emails were received by the company.
3. Test, at the \(10 \%\) level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men’s Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly.

1. A cadet fires shots at a target at distances ranging from 25 m to 90 m . The probability of hitting the target with a single shot is \(p\). When firing from a distance \(d \mathrm {~m} , p = \frac { 3 } { 200 } ( 90 - d )\). Each shot is fired independently.

The cadet fires 10 shots from a distance of 40 m .

1. 1. Find the probability that exactly 6 shots hit the target.
   2. Find the probability that at least 8 shots hit the target. The cadet fires 20 shots from a distance of \(x \mathrm {~m}\).
2. Find, to the nearest integer, the value of \(x\) if the cadet has an \(80 \%\) chance of hitting the target at least once. The cadet fires 100 shots from 25 m .
3. Using a suitable approximation, estimate the probability that at least 95 of these shots hit the target.

5. (a) State the conditions under which the normal distribution may be used as an approximation to the binomial distribution. A company sells seeds and claims that 55\% of its pea seeds germinate.
(b) Write down a reason why the company should not justify their claim by testing all the pea seeds they produce. To test the company's claim, a random sample of 220 pea seeds was planted.
(c) State the hypotheses for a two-tailed test of the company's claim. Given that 135 of the 220 pea seeds germinated,
(d) use a normal approximation to test, at the \(5 \%\) level of significance, whether or not the company’s claim is justified.

6. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { 9 } & 0 \leqslant x \leqslant 1
\frac { 2 } { 9 } & 1 < x < 4
\frac { 2 } { 3 } - \frac { x } { 9 } & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{array} \right.$$

1. Find \(\mathrm { E } ( X )\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
3. Find the median of \(X\).
4. Describe the skewness. Give a reason for your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{caaa0133-5b13-4ca7-8e65-8543327c33fd-12_104_61_2412_1884}